As far as I am concerned, given m >= n, the result of QR(A_m,n) would be V_m,n T_n,n and R_m,n.
In LAPACK Working Note # 191, the DTSQRT is described below:
( Rkk) (Rkk)
( ) -----> (Vik, Tik, Rkk) = QR( )
( Aik ) (Aik )
Since the coupling of Rkk and Aik is a (2b)X(b) block, I think m = 2b and n = b here. Therefore I think Vik and Rkk are no longer bXb blocks but (2b)X(b). How can these blocks fit into original matrix? For Rkk, maybe we can just ignore the lower bXb zero block. But I don't think we can do the same thing for Vik.